Animating wire/cable/rope wrapping around a cylinder
This is a somewhat similar answer to @moonboots', perhaps expressed in a different way.

Using the shipped add-on Add Curve: Extra Objects, create an Archimedean Spiral.
Create your rope as a straight subdivided cylinder of some kind: it could be an array of segments, or simply another curve. Mine is long along X.
Use a Curve modifier to deform your rope along the spiral.
As you move the rope along its deforming axis (my X), it will wind itself around the spiral, and, when it runs out of curve, project itself from the end in the direction of the spiral's last segment. With (T)interpolation set to 'Linear' in the timeline, keyframe the rope's start and end positions, for a whole number of turns.
Now, the trick is to keep the fixed end of the rope stationary in the length-dimension of the spiral. Again, with interpolation set to 'Linear', keyframe the spiral's translation the other way, to compensate for the movement  of the tail of the rope along it. Neither the spiral nor the rope rotate.
Create a spool that does rotate, inside the spiral; keyframe its rotation to sync with the rope's progress around the spiral. You know how many turns the rope has travelled between start and end frames; just keyframe that number of turns into the spool.